The need for formalism in arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1888, Richard Dedekind proposed a collection of axioms about the numbers, and in 1889 Peano published a more precisely formulated version of them as a collection of axioms in his book, The principles of arithmetic presented by a new method (Arithmetices principia, nova methodo exposita).

The Peano axioms contain three types of statements. The first four statements are general statements about equality; in modern treatments these are often considered axioms of pure logic. The next four axioms are first-order statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is a second order statement of the principle of mathematical induction over the natural numbers. A weaker first-order system called Peano arithmetic is obtained by replacing this second-order induction axiom with a first-order axiom schema.

The axioms

When Peano formulated his axioms, the language of mathematical logic was in its infancy. The system of logical notation he created to present the axioms did not prove to be popular, although it was the genesis of the modern notation for set membership (∈ from Peano's ε) and implication (⊃ from Peano's reversed 'C'). Peano maintained a clear distinction between mathematical and logical symbols, which was not yet common in mathematics; such a separation had first been introduced in the Begriffsschrift by Gotlob Frege, published in 1879. Peano was unaware of Frege's work and independently recreated his logical apparatus based on the work of Boole and Schröder.

The Peano axioms define the properties of natural numbers, usually represented as a setN or mathbb{N}. The first four axioms describe the equalityrelation.

For all natural numbers x and y, if x = y, then y = x. That is, equality is symmetric.

For all natural numbers x, y and z, if x = y and y = z, then x = z. That is, equality is transitive.

For all a and b, if a is a natural number and a = b, then b is also a natural number. That is, the natural numbers are closed under equality.

The remaining axioms define the properties of the natural numbers. The constant 0 is assumed to be a natural number, and the naturals are assumed to be closed under a "successor" functionS.

0 is a natural number.

For every natural number n, S(n) is a natural number.

Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number. This choice is arbitrary, as axiom 5 does not endow the constant 0 with any additional properties. However, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0. Axioms 5 and 6 define a unary representation of the natural numbers: the number 1 is S(0), 2 is S(S(0)) (= S(1)), and, in general, any natural number n is Sn(0). The next two axioms define the properties of this representation.

For every natural number n, S(n) ≠ 0. That is, there is no natural number whose successor is 0.

For all natural numbers m and n, if S(m) = S(n), then m = n. That is, S is an injection.

These two axioms together imply that the set of natural numbers is infinite, because it contains at least the infinite subset { 0, S(0), S(S(0)), … }, each element of which differs from the rest. The final axiom, sometimes called the axiom of induction, is a method of reasoning about all natural numbers; it is the only second order axiom.

for a, b in N , a le b if and only if there exists c in N such that a + c = b .

This relation is stable under addition and multiplication: for a, b, c in N , if a ≤ b, then:

a + c ≤ b + c, and

a · c ≤ b · c.

Thus, the structure (N, +, ·, 1, 0, ≤) is an ordered semiring; because there is no natural number between 0 and 1, it is a discrete ordered semiring. The axiom of induction is sometimes stated in the following strong form, making use of the ≤ order:

* for every n, k ∈ N, if k ≤ n implies φ(k) is true, then φ(S(n)) is true,

then for every n ∈ N, φ(n) is true.

This form of the induction axiom is a simple consequence of the standard formulation, but is often better suited for reasoning about the ≤ order. For example, to show that the naturals are well-ordered—every nonemptysubset of N has a least element—one can reason as follows. Let a nonempty X ⊆ N be given and assume X has no least element.

Because 0 is the least element of N, it must be that 0 ∉ X.

For any n ∈ N, suppose for every k ≤ n, k ∉ X. Then S(n) ∉ X, for otherwise it would be the least element of X.

Thus, by the strong induction principle, for every n ∈ N, n ∉ X. Thus, X ∩ N = ∅, which contradictsX being a nonempty subset of N.Thus X has a least element.

Models

A model of the Peano axioms is a triple (N, 0, S), where N an infinite set, 0 ∈ N and S : N → N satisfies the axioms above. Dedekind proved in his 1888 book, What are numbers and what should they be (Was sind und was sollen die Zahlen) that any two models of the Peano axioms are isomorphic: given two models (NA, 0A, SA) and (NB, 0B, SB) of the Peano axioms, the homomorphismf : NA → NB defined as

begin{align}

f(0_A) &= 0_B
f(S_A (n)) &= S_B (f (n))
end{align}
is a bijection. The Peano axioms are thus categorical; this is not the case with any first-order reformulation of the Peano axioms, however.

First-order theory of arithmetic

First-order theories are often better than second order theories for model or proof theoretic analysis. All but the ninth axiom (the induction axiom) are statements in first-order logic. The arithmetical operations of addition and multiplication and the order relation can also be defined using first-order axioms. The second-order axiom of induction can be transformed into a weaker first-order induction schema; the first eight of Peano's axioms together with the first-order induction schema
form a first-order axiomatization of arithmetic called Peano arithmetic (PA).

The induction schema consists of a countably infinite set of axioms. For each formula φ(x,y1,...,yk) in the language of Peano arithmetic, the first-order induction axiom for φ is the sentence

where bar{y} is an abbreviation for y1,...,yk. The first-order induction schema includes every instance of the first-order induction axiom, that is, it includes the induction axiom for every formula φ.

This schema avoids quantification over sets of natural numbers, which is impossible in first-order logic. For instance, it is not possible in first-order logic to say that any set of natural numbers containing 0 and closed under successor is the entire set of natural numbers. What can be expressed is that any definable set of natural numbers has this property. Because it is not possible to quantify over definable subsets explicitly with a single axiom, the induction schema includes one instance of the induction axiom for every definition of a subset of the naturals.

Equivalent axiomatizations

There are many different, but equivalent, axiomatizations of Peano arithmetic. While some axiomatizations, such as the one just described, only describe the successor operation, other axiomatizations directly describe the arithmetical operations. One such axiomatization begins with the following axioms that describe a discrete ordered semiring.

The system of logic defined by these axioms is known as PA–; PA is obtained by adding the first-order induction schema. An important property of PA– is that any structure M satisfying this theory has an initial segment (ordered by ≤) isomorphic to N. Elements of MN are known as nonstandard elements.

Nonstandard models

Although the usual natural numbers satisfy the axioms of PA, there are other non-standard models as well; the compactness theorem implies that the existence of nonstandard elements cannot be excluded in first-order logic. The upward Löwenheim-Skolem theorem shows that there are nonstandard models of PA of all infinite cardinalities. This is not the case for the original (second-order) Peano axioms, which have only one model, up to isomorphism. This illustrates one way the first-order system PA is weaker than the second-order Peano axioms.

When interpreted as a proof within a first-order set theory, such as ZFC, Dedekind's categoricity proof for PA shows that each model of set theory has a unique model of the Peano axioms, up to isomorphism, that embeds as an initial segment of all other models of PA contained within that model of set theory. In the standard model of set theory, this smallest model of PA is the standard model of PA; however, in a nonstandard model of set theory, it may be a nonstandard model of PA. This situation cannot be avoided with any first-order formalization of set theory.

It is natural to ask whether a countable nonstandard model can be explicitly constructed. It is possible to explicitly describe the order type of any countable nonstandard model: it is always ω + η (ω* + ω), which can be visualized as a copy of the natural numbers followed by a dense linear ordering of copies of the integers. However, a theorem by Stanley Tennenbaum, proved in 1959, shows that there is no countable nonstandard model of PA in which either the addition or multiplication operation is computable. This result shows it is difficult to be completely explicit in describing the addition and multiplication operations of a countable nonstandard model of PA.

Set-theoretic models

The Peano axioms can be derived from set theoretic constructions of the natural numbers and axioms of set theory such as the ZF. The standard construction of the naturals, due to John von Neumann, starts from a definition of 0 as the empty set, ∅, and an operator s on sets defined as:

s(a) = a ∪ { a }.

The set of natural numbers N is defined as the intersection of all sets closed under s that contain the empty set. Each natural number is equal (as a set) to the set of natural numbers less than it:

Peano arithmetic is equiconsistent with several weak systems of set theory. One such system is ZFC with the axiom of infinity replaced by its negation. Another such system consists of general set theory (extensionality, existence of null set, and the axiom of adjunction), augmented by an axiom schema stating that a property that holds for the empty set and holds of an adjunction whenever it holds of the adjunct must hold for all sets.

Interpretation in category theory

A model of the Peano axioms can also be constructed using category theory. Let C be a category with initial object 1C, and define the category of pointed unary systems, US1(C) as follows:

The objects of US1(C) are triples (X, 0X, SX) where X is an object of C, and 0X : 1C → X and SX : X → X are C-morphisms.

Then C is said to satisfy the Dedekind-Peano axioms if US1(C) has an initial object; this initial object is known as a natural number object in C. If (N, 0, S) is this initial object, and (X, 0X, SX) is any other object, then the unique map u : (N, 0, S) → (X, 0X, SX) is such that

Consistency

When the Peano axioms were first proposed, Bertrand Russell and others agreed that these axioms implicitly defined what we mean by a "natural number". Henri Poincaré was more cautious, saying they only defined natural numbers if they were consistent; if there is a proof that starts from just these axioms and derives a contradiction such as 0 = 1, then the axioms are inconsistent, and don't define anything. In 1900, David Hilbert posed the problem of proving their consistency using only finitistic methods as the second of his twenty-three problems. In 1931, Kurt Gödel proved his second incompleteness theorem, which shows that such a consistency proof cannot be formalized within Peano arithmetic itself.

Although it is widely claimed that Gödel's theorem rules out the possibility of a finitistic consistency proof for Peano arithmetic, this depends on exactly what one means by a finitistic proof. Gödel himself pointed out the possibility of giving a finitistic consistency proof of Peano arithmetic or stronger systems by using finitistic methods that are not formalizable in Peano arithmetic, and in 1958 Gödel published a method for proving the consistency of arithmetic using type theory. In 1936, Gerhard Gentzen gave a proof of the consistency of Peano's axioms, using transfinite induction up to an ordinal called ε0. Gentzen explained: "The aim of the present paper is to prove the consistency of elementary number theory or, rather, to reduce the question of consistency to certain fundamental principles". Gentzen's proof is arguably finitistic, since the transfinite ordinal ε0 can be encoded in terms of finite objects (for example, as a Turing machine describing a suitable order on the integers). Whether or not Gentzen's proof meets the requirements Hilbert envisioned is unclear: there is no generally accepted definition of exactly what is meant by a finitistic proof, and Hilbert himself never gave a precise definition.

The vast majority of contemporary mathematicians believe that Peano's axioms are consistent, relying either on intuition or the acceptance of a consistency proof such as Gentzen's proof. The small number of mathematicians who advocate ultrafinitism reject Peano's axioms because the axioms require an infinite set of natural numbers.

Jean van Heijenoort, ed. (1967, 1976 3rd printing with corrections). From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931. 3rd, Cambridge, Mass: Harvard University Press. ISBN 0-674-32449-8 (pbk.). Contains translations of the following two papers, with valuable commentary:

Richard Dedekind, 1890, "Letter to Keferstein." pp. 98-103. On p. 100, he restates and defends his axioms of 1888.

Guiseppe Peano, 1889. Arithmetices principia, nova methodo exposita (The principles of arithmetic, presented by a new method), pp. 83-97. An excerpt of the treatise where Peano first presented his axioms, and recursively defined arithmetical operations.